Some general results in the theory of crystallographic slip

  • P. M. Naghdi
  • A. R. Srinivasa
Original Papers


Crystallographic slip of a Bravais lattice is analyzed utilizing the main results of a recently constructed theory of structured solids, where explicit account is taken of the influence of dislocation density identified in terms of Curl of plastic deformationG p . In the present paper, the scope of the subject is enlarged to also include defects (other than dislocations) such as substitutional impurities and vacancies and it is shown that these point defects may also be characterized in terms of the plastic deformation fieldG p . Several general results pertaining to the kinematics and kinetics of Crystallographic slip are proved within the scope of an appropriate constraint theory suitable for Crystallographic slip; the latter is motivated by the well-known basic mechanism of Crystallographic slip that constrains the admissible modes of plastic deformation. The constraint responses (or forces) that are necessary to maintain the active slip systems, as well as the conditions for the transitions between the slip systems, are determined. In spite of the nature of the assumption pertaining to the mechanism of Crystallographic slip on distinct slip systems, it is shown that the yield surface does not necessarily exhibit sharp corners. Instead, the shape of the yield surface is in the form of hyperplanes joined by round corners. In fact, the presence of sharp corners is mainly a result of the use of a special set of constitutive assumptions. The predictive capability of the theoretical results is further illustrated by using a two-dimensional crystal subjected to simple shear. The effect of the initial dislocation density on the response of the sheared-crystal is studied by carrying out detailed calculations for two substantially different initial dislocation densities. The calculations show that while the response of the crystal is sensitive to the initial dislocation density in the early stages of deformation, its influence diminishes with progressively larger deformations. Furthermore, the crystal exhibits a well-defined shear band which evolves naturally due to the presence of initial dislocation distribution and is easily visible at large deformations.


Shear Band Slip System Yield Surface Simple Shear Sharp Corner 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bravais, M. D.,Mé moire sur les systèmes formé par points distribués régulièrement sur un plan ou dans l'espace. J. Ecole Polytech., Cah. 33, T. XIX, Paris, pp. 1–128, 1850. [Translation by A. J. Shaler.On the systems formed by points regularly distributed on a plane or in space. American Crystallographic Ass., Monograph 3 (1949).]Google Scholar
  2. [2]
    Ewing, J. A. and Rosenhain, W.,The crystalline structure of metals. Phil. Trans. R. Soc. London193, 353–375 (1990).Google Scholar
  3. [3]
    Taylor, G. I. and Elam, C. F.,The distortion of an aluminum crystal during a tensile test. Proc. R. Soc. London A102, 643–667 (1923).Google Scholar
  4. [4]
    Taylor, G. I. and Elam, C. F.,The plastic extension and fracture of aluminum crystals. Proc. R. Soc. London A108, 28–51 (1925).Google Scholar
  5. [5]
    Taylor, G. I. and Elam, C. F.,The distortion of iron crystals. Proc. R. Soc. London A112, 337–361. (1926).Google Scholar
  6. [6]
    Piercy, G. R., Cahn, R. W. and Cottrell, A. H.,A study of primary and conjugate slip in crystals of alpha-brass. Acta Metall.3, 331–338 (1955).Google Scholar
  7. [7]
    Kocks, U. F.,Polyslip in polycrystals. Acta Metall.6, 85–94 (1958).Google Scholar
  8. [8]
    Taylor, G. I.,The mechanism of plastic deformation of crystals. Part I. Theoretical. Proc. R. Soc. London A145, 332–387 (1934).Google Scholar
  9. [9]
    Van Bueren, H. G.,Imperfections in Crystals, 2nd Ed., North-Holland Publ. Co., Amsterdam 1961.Google Scholar
  10. [10]
    Asaro, R. J., Micromechanics of crystals and polycrystals. InAdvances in Applied Mechanics, Vol. 23, Academic Press, pp. 1–115, 1982.Google Scholar
  11. [11]
    Naghdi, P. M.,A critical review of the state of finite plasticity. J. Appl. Math. Phys. (ZAMP)41, 315–394 (1990).Google Scholar
  12. [12]
    Havner, K. S.,Finite Plastic Deformation of Crystalline Solids, Cambridge Univ. Press 1992.Google Scholar
  13. [13]
    Oden, J. T. and Carey, G. F.,Finite Elements: Special Problems in Solid Mechanics, Vol. 5, Prentice-Hall Inc., Englewood Cliffs, N.J. 1984.Google Scholar
  14. [14]
    Hirth, J. P. and Lothe, J.,Theory of Dislocations, 2nd Ed., John Wiley & Sons 1982.Google Scholar
  15. [15]
    Eshelby, J. D., The continuum theory of lattice defects. InSolid State Physics, Vol. 3 (edited by F. Scitz and D. Turbull), Academic Press, pp. 79–144, 1956.Google Scholar
  16. [16]
    Bilby, B. A.,Continuous distributions of dislocations. InProgress in Solid Mechanics, Vol. 1 (edited by I. N. Sneddon and R. Hill), North-Holland Publ. Co., pp. 331–398, 1960.Google Scholar
  17. [17]
    Naghdi, P. M. and Srinivasa, A. R.,A dynamical theory of structured solids. Part I. Basic developments. Phil. Trans. R. Soc. London A345, 425–458 (1993).Google Scholar
  18. [18]
    Naghdi, P. M. and Srinivasa, A. R.,A dynamical theory of structured solids. Part II. Special constitutive equations and special cases of the theory. Phil. Trans. R. Soc. London A345, 459–476 (1993).Google Scholar
  19. [19]
    Naghdi, P. M. and Srinivasa, A. R.,On the characterization of dislocations and their influence on plastic deformation in single crystals. Int. J. Engng. Sci. (1993), in press.Google Scholar
  20. [20]
    Nabarro, F. N.,Theory of Crystal Dislocations, Dover Publ., New York. [This is a slightly corrected version of the book first published in 1967 by Oxford University Press.]|Google Scholar
  21. [21]
    Guinier, A. and Jullien, R.,The solid state: from superconductors to superalloys (translated from the FrenchLa matière à l'état solide: des supraconducteurs aux superalliages by W. J. Duffin). IUCr. Texts of Crystallography 1, Oxford University Press, New York 1989.Google Scholar
  22. [22]
    Hertzberg, R. W.,Deformation and Fracture Mechanics of Engineering Materials, John Wiley & Sons, New York 1989.Google Scholar
  23. [23]
    Guinier, A.,The Structure of Matter — From the Blue Sky to Liquid Crystals (translated from the 1980 French edition by W. J. Duffin), Edward Arnold, London 1984.Google Scholar
  24. [24]
    Barrett, C. S., Crystal structure of metals. InMetals Handbook, 8th Ed., ASM, Metals Park, Ohio, pp. 233–250, 1973.Google Scholar
  25. [25]
    Green, A. E., Naghdi, P. M. and Trapp, J. A.,Thermodynamics of a continuum with internal constraints. Int. J. Engng. Sci.8, 891–908 (1970).Google Scholar
  26. [26]
    Green, A. E. and Naghdi, P. M.,On thermodynamics and the nature of the second law. Proc. R. Soc. London A357, 253–270 (1977).Google Scholar
  27. [27]
    Green, A. E. and Naghdi, P. M.,A demonstration of consistency of an entropy balance with balance of energy. J. Appl. Math. Phys. (ZAMP)42, 159–168 (1991).Google Scholar
  28. [28]
    Franciosi, P. and Zaoui, A.,multiple slip in FCC crystals: a theoretical approach compared with experimental data. Acta Metallurgica et Materiala30, 1627–1633 (1982).Google Scholar
  29. [29]
    Tanner, B. K. and Bowen, D. K.,Synchrotron X-ray topography. Materials Science Reports8, 369–407 (1992).Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • P. M. Naghdi
    • 1
  • A. R. Srinivasa
    • 1
  1. 1.Dept of Mechanical EngineeringUniversity of CaliforniaBerkeleyUSA

Personalised recommendations